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We study properties of two resampling scenarios: Conditional Randomisationand Conditional Permutation schemes, which are relevant for testing conditionalindependence of discrete random variables $X$ and $Y$ given a random variable$Z$. Namely, we investigate asymptotic behaviour of estimates of a vector ofprobabilities in such settings, establish their asymptotic normality andordering between asymptotic covariance matrices. The results are used to deriveasymptotic distributions of the empirical Conditional Mutual Information inthose set-ups. Somewhat unexpectedly, the distributions coincide for the twoscenarios, despite differences in the asymptotic distributions of the estimatesof probabilities. We also prove validity of permutation p-values for theConditional Permutation scheme. The above results justify consideration ofconditional independence tests based on resampled p-values and on theasymptotic chi-square distribution with an adjusted number of degrees offreedom. We show in numerical experiments that when the ratio of the samplesize to the number of possible values of the triple exceeds 0.5, the test basedon the asymptotic distribution with the adjustment made on a limited number ofpermutations is a viable alternative to the exact test for both the ConditionalPermutation and the Conditional Randomisation scenarios. Moreover, there is nosignificant difference between the performance of exact tests for ConditionalPermutation and Randomisation schemes, the latter requiring knowledge ofconditional distribution of $X$ given $Z$, and the same conclusion is true forboth adaptive tests.

Analysis of Conditional Randomisation and Permutation schemes with application to conditional independence testing